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A restaurant's revenue history for the first 10 years in business is modeled by the function:

[tex]\[ R=0.0001\left(-t^4+12 t^3-77 t^2+600 t+13,050\right) \][/tex]

where [tex]\( t \)[/tex] is the number of years since the restaurant opened and [tex]\( R \)[/tex] is annual revenue in millions of dollars.

In which year(s) did the restaurant's revenue equal \[tex]$1.5 million?

The restaurant's revenue equals \$[/tex]1.5 million for the first time in year _______ and again in year _______.



Answer :

GinnyAnswer
Let's analyze the problem step by step:

### Objective:

We need to determine the years [tex]\( t \)[/tex] (since the restaurant opened) when the restaurant's revenue [tex]\( R(t) \)[/tex] is equal to [tex]$1.5$[/tex] million dollars.

### Given Revenue Function:

[tex]\[ R(t) = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) \][/tex]

### Revenue Target:

The target revenue is [tex]$1.5$[/tex] million, which we can write as:

[tex]\[ R(t) = 1,500,000 \][/tex]

### Setting Up the Equation:

We equate the revenue function to [tex]$1.5$[/tex] million.

[tex]\[ 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,050) = 1,500,000 \][/tex]

To simplify calculations, we can multiply both sides of the equation by [tex]$10,000$[/tex] (this cancels out the [tex]$0.0001$[/tex] factor):

[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 = 15,000,000 \][/tex]

### Rearrange the Equation:

We rearrange the equation to set it to zero, as this forms a polynomial that we can solve for roots:

[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t + 13,050 - 15,000,000 = 0 \][/tex]

This further simplifies to:

[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]

### Solving the Polynomial:

To find the values of [tex]\( t \)[/tex], we solve the polynomial equation:

[tex]\[ -t^4 + 12t^3 - 77t^2 + 600t - 14,986,950 = 0 \][/tex]

### Results:

Upon solving this polynomial equation, we find that there are no real and positive solutions for [tex]\( t \)[/tex].

### Conclusion:

Since there are no positive, real solutions for [tex]\( t \)[/tex], the restaurant's revenue never reaches exactly [tex]$1.5$[/tex] million at any point during the first 10 years.

So, the answer is:

The restaurant's revenue does not equal [tex]$1.5$[/tex] million in any year within the timeframe given.

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